Theorem bd0 16764

Description: A formula equivalent to a bounded one is bounded. See also bd0r 16765. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bd0.min	⊢ BOUNDED 𝜑
bd0.maj	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bd0	⊢ BOUNDED 𝜓

Proof of Theorem bd0

Step	Hyp	Ref	Expression
1		bd0.min	. 2 ⊢ BOUNDED 𝜑
2		bd0.maj	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	ax-bd0 16753	. 2 ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
4	1, 3	ax-mp 5	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 105  BOUNDED wbd 16752

This theorem was proved from axioms:  ax-mp 5  ax-bd0 16753

This theorem is referenced by:  bd0r  16765  bdth  16771  bdnth  16774  bdnthALT  16775  bdph  16790  bdsbc  16798  bdsnss  16813  bdcint  16817  bdeqsuc  16821  bdcriota  16823  bj-axun2  16855